We want to obtain bounds for \(|\hat{{\nabla }}^pg_m(t)|_{\hat{g}}\)  for any \(p\) . First of all (see [1]} for details), \(h^{-1}_m\hat{{\nabla }}h_m = {\nabla }^m h_m h^{-1}_m = {\nabla }^m - \hat{{\nabla }} = g^{-1}\hat{{\nabla }}g\) , where \(h_m\)  is a relative endomorphism \((h_m)^a_b = \hat{g}^{a\bar{s}}(g_m)_{b\bar{s}}\) . Next, since we proved the equivalence for \(\hat{g}\)  and \(g_m\) , the norms on tensors, induced by these metrics are equivalent (see [Lemma 3.13][2]} for the proof).
